#6682817

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2023 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,22)(15,19,26,33)(16,20,27,34)(17,21,29,36)(23,30,25,32)(24,31,39,43)(28,35,40,44)(37,41,38,42);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,24)(17,23)(18,26)(20,31)(21,30)(22,33)(25,29)(27,39)(28,37)(32,36)(34,43)(35,41)(38,40)(42,44);(1,2,3,4)(15,23)(17,29)(19,30)(21,36)(24,37)(25,26)(28,40)(31,41)(32,33)(35,44)(38,39)(42,43);(6,10)(9,11)(13,16)(14,20)(15,24)(17,28)(18,27)(19,31)(21,35)(22,34)(23,37)(25,38)(26,39)(29,40)(30,41)(32,42)(33,43)(36,44);(7,12)(9,11)(15,23)(16,27)(17,29)(19,30)(20,34)(21,36)(24,38)(25,26)(31,42)(32,33)(37,39)(41,43)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6590013);